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ACS Photonics

Constructing “Designer Atoms” via Resonant Graphene-Induced Lamb Shifts

Cyuan-Han Chang, Nicholas Rivera, John D. Joannopoulos, Marin Soljacic A, Ido E. Kaminer

Published Sep 12, 2017• 17 citations
Publisher

Abstract

The properties of an electron in an atom or molecule are not fixed; rather they are a function of the optical environment of the emitter. Not only is the spontaneous emission a function of the optical environment, but also the underlying wave functions and energy levels, which are modified by the potential induced by quantum fluctuations of the electromagnetic field. In free space, this modification of atomic levels and wave functions is very weak and generally hard to observe due to the prevalence of other perturbations like fine structure. Here, we explore the possibility of highly tailorable electronic structure by exploiting large Lamb shifts in tunable electromagnetic environments such as graphene, whose optical properties are dynamically controlled via doping. The Fermi energy can be chosen so that the Lamb shift is very weak, but it can also be chosen so that the shifts become more prominent than the fine structure of the atom and even potentially the Coulomb interaction with the nucleus. Potential implications of this idea include being able to electronically shift an unfavorable emitter structure into a favorable one, a new approach to probe near-field physics in fluorescence, and a way to access regimes of physics where vacuum fluctuations are not a weak perturbation but rather the dominant physics.

DOI
10.1021/acsphotonics.7b00731

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The properties of an electron in an atom or molecule are not fixed; rather they are a function of the optical environment of the emitter. Not only is the spontaneous emission a function of the optical environment, but also the underlying wave functions and energy levels, which are modified by the potential induced by quantum fluctuations of the electromagnetic field. In free space, this modification of atomic levels and wave functions is very weak and generally hard to observe due to the prevalence of other perturbations like fine structure. Here, we explore the possibility of highly tailorable electronic structure by exploiting large Lamb shifts in tunable electromagnetic environments such as graphene, whose optical properties are dynamically controlled via doping. The Fermi energy can be chosen so that the Lamb shift is very weak, but it can also be chosen so that the shifts become more prominent than the fine structure of the atom and even potentially the Coulomb interaction with the nucleus. Potential implications of this idea include being able to electronically shift an unfavorable emitter structure into a favorable one, a new approach to probe near-field physics in fluorescence, and a way to access regimes of physics where vacuum fluctuations are not a weak perturbation but rather the dominant physics.
An atomic electron, in the vacuum of photons, can emit and reabsorb photons of any frequency. Such processes give rise to well known radiative corrections (Lamb—shift, g — 2, natural width of excited states...). We review in this paper similar effects which can be induced by the interaction of an atomic electron with an incident electromagnetic wave, and we try to get some physical insight in radiative processes by comparing these two types of « spontaneous » and « induced » effects. Optical pumping methods have been very useful for studying how atoms are perturbed by optical or RF photons. The Zeeman sublevels of an atomic ground state are broadened and shifted by a resonant or quasiresonant optical irradiation. The corresponding light-shifts (or « Lamp-shifts » as A. Kastler called them) are due to virtual absorptions and reemissions of incident photons. Another example of an induced effect is the modification of the magnetic properties of an atomic level due to a high frequency non resonant RF irradiation. The Larmor precession is slowed down and becomes anisotropic. The two previous examples correspond to two situations leading to simple calculations. For a resonant or quasiresonant excitation, a single atomic transition needs to be considered. Non perturbative treatments can be given which include the effect of the atom field coupling to all orders. For example, the « dressed atom » approach allows a unified treatment of various effects such as light shifts, Autler-Townes doublets resonance fluorescence triplets... On the other hand, for a high frequency irradiation, all atomic transitions must be included. It is possible in this case to derive an effective hamiltonian describing, to lowest order in the fine structure constant α, how electronic energy levels are perturbed in the presence of N photons ω. The remaining part of the paper is devoted to the analysis of such a situation. The terms of the effective hamiltonian proportional to the number N of incident photons correspond to the induced effects. They can be fully interpreted in semiclassical terms and they describe how the slow motion of the electron in the external static fields is modified by the high frequency vibration in the incident monochromatic wave. For example, the vibrating electron averages the static potential over a finite length, the angular oscillation of the spin reduces the effective magnetic moment. The remaining terms, which are independent of N, exist even in the vacuum of photons (N = 0) and describe the contribution of the mode ω to spontaneous radiative corrections. It is possible to split them into two parts. The first one has the same structure as the N-dependent terms, with N replaced by 1/2. It describes the corrections due to the vibration of the electron in vacuum fluctuations which have a spectral power density ħω/2 per mode. The second part represents the effect of radiation reaction, i.e. the interaction of the electron with its self field. Such an interpretation is confirmed by a Heisenberg equations approach where the total force acting upon the electron is split into two physical (i.e. hermitian) parts representing respectively the effect of the vacuum « free » field and the effect of the « source » field produced by the electron itself. These considerations are finally applied to the problem of the positive sign of g - 2. The g-factor can be written as 2 ωL/ωc, where ω L is the Larmor frequency of the spin and ωc the cyclotron frequency of the charge for an electron in a weak static field B 0. In the absence of radiative corrections, g = 2, and ω L = ωc. To lowest order in 1/c (order zero), radiation reaction slows down ωc but not ωL (a charge is more coupled to its self field than a magnetic moment). The effect of vacuum fluctuations appears only at order 2 in 1/c and corresponds to relativistic and magnetic effects which slow down both ωL and ωc. Therefore, it appears that, in the non relativistic domain, the main effect is a reduction of ωc by radiation reaction and this explains why 2 ωL/ωc becomes larger than 2. A complete relativistic calculation (to all orders in 1/ c, but to lowest order in α) does not change this conclusion since the contribution of non relativistic modes (ħω < mc2) is predominant in the integral giving g — 2.
Electrically tunable graphene plasmons are anticipated to enable strong light-matter interactions with resonant quantum emitters. However, plasmon resonances in graphene are typically limited to infrared frequencies, below those of optical excitations in robust quantum light sources and many biologically interesting molecules. Here we propose to utilize near fields generated by the plasmon-assisted nonlinear optical response of nanostructured graphene to resonantly couple with proximal quantum emitters operating in the near infrared. We show that the nonlinear near field produced by a graphene nanodisk can strongly excite and coherently control quantum states in two- and three-level atomic systems when the third harmonic of its plasmon resonance is tuned to a particular electronic transition. In the present scheme, emitter and plasmon resonances are nondegenerate, circumventing strong enhancement of spontaneous emission. We envision potential applications for the proposed nonlinear plasmonic coupling scheme in sensing and temporal quantum control.
By engineering the electromagnetic vacuum field, the induced Casimir-Polder shift (also known as Lamb shift) and spontaneous emission rates of individual atomic levels can be controlled. When the strength of these effects becomes comparable to the energy difference between two previously uncoupled atomic states, an environment-induced interaction between these states appears after tracing over the environment. This interaction has been previously studied for degenerate levels and simple geometries involving infinite, perfectly conducting half-spaces or free space. Here, we generalize these studies by developing a convenient description that permits the analysis of these non-diagonal perturbations to the atomic Hamiltonian in terms of an accurate non-Hermitian Hamiltonian. Applying this theory to a hydrogen atom close to a dielectric nanoparticle, we show strong vacuum-field-induced state mixing that leads to drastic modifications in both the energies and decay rates compared to conventional diagonal perturbation theory. In particular, contrary to the expected Purcell enhancement, we find a surprising decrease of decay rates within a considerable range of atom-nanoparticle separations. Furthermore, we quantify the large degree of mixing of the unperturbed eigenstates due to the non-diagonal perturbation. Our work opens new quantum state manipulation possibilities in emitters with closely spaced energy levels.
Abstract Quantum fluctuations of the electromagnetic field are known to produce the atomic Lamb shift. We here reveal their iconic signature in semiconductor physics, through the blue-shift they produce to optically bright excitons, thus lifting the energy of these excitons above their dark counterparts. The electromagnetic field here acts in its full complexity: in addition to its longitudinal part via inter band virtual Coulomb processes, the transverse part —which has been missed up to now— also acts via resonant and nonresonant virtual photons . These two parts beautifully combine to produce a bright exciton blue-shift independent of the exciton wave vector direction. Our work readily leads to a striking prediction: long-lived excitons must have a small bright-dark splitting. Although the analogy between exciton and hydrogen atom could lead us to see the bright exciton shift as a Lamb shift, this is not fully so: the Lamb shift comes entirely from virtual photons, whereas the Coulomb interaction also contributes to the exciton shift through the so-called “electron-hole exchange”.
Combining the quantum optical properties of single-photon emitters with the strong near-field interactions available in nanophotonic and plasmonic systems is a powerful way of creating quantum manipulation and metrological functionalities. The ability to actively and dynamically modulate emitter-environment interactions is of particular interest in this regard. While thermal, mechanical and optical modulation have been demonstrated, electrical modulation has remained an outstanding challenge. Here we realize fast, all-electrical modulation of the near-field interactions between a nanolayer of erbium emitters and graphene, by in-situ tuning the Fermi energy of graphene. We demonstrate strong interactions with a >1000-fold increased decay rate for ~25% of the emitters, and electrically modulate these interactions with frequencies up to 300 kHz - orders of magnitude faster than the emitter's radiative decay (~100 Hz). This constitutes an enabling platform for integrated quantum technologies, opening routes to quantum entanglement generation by collective plasmon emission or photon emission with controlled waveform.